Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back

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1 Navigating Cntral Path with Elctrical Flows: from Flows to Matchings, and Back Alksandr Mądry EPFL Abstract 10 W prsnt an Õ(m 7 ) = Õ(m1.43 )-tim 1 algorithm for th maximum s-t flow and th minimum s-t cut problms in dirctd graphs with unit capacitis. This is th first improvmnt ovr th spars-graph cas of th long-standing O(m min{ m, n 2/3 }) running tim bound du to Evn and Tarjan [ET75] and Karzanov [Kar73]. By wll-known rductions, this also stablishs 10 an Õ(m 7 )-tim algorithm for th maximum-cardinality bipartit matching problm. That, in turn, givs an improvmnt ovr th clbratd O(m n) running tim bound of Hopcroft and Karp [HK73] and Karzanov [Kar73] whnvr th input graph is sufficintly spars. At a vry high lvl, our rsults stm from acquiring a dpr undrstanding of intrior-point mthods a powrful tool in convx optimization in th contxt of flow problms, as wll as, utilizing crtain intrplay btwn maximum flows and bipartit matchings. Th cor of our approach compriss a primal-dual algorithm for (nar-)prfct bipartit b- matching problm. This algorithm is inspird by path-following intrior-point mthods and mploys lctrical flow computations to gradually improv th quality of maintaind solution by advancing it toward (nar-)optimality along so-calld cntral path. To analyz this procss, w stablish a formal connction that tis its convrgnc rat to th structur of corrsponding lctrical flows. Thn, w xploit that connction to obtain a convrgnc guarant for our algorithm that improvs upon th wll-known barrir of Ω( m) itrations corrsponding to th gnric worst-cas prformanc bounds for intrior-point-mthod-basd algorithms. This improvmnt is basd on rfining crtain insights into bhavior of lctrical flows that stm from th work of Christiano t al. [CKM + 11] and combining thm with a nw tchniqu for prconditioning primal-dual solutions. Th final ingrdint of our approach is a simpl rduction of th maximum s-t flow problm to th bipartit b-matching problm. This rduction is thn composd with th rcnt sublinar-tim algorithm for finding prfct matchings in rgular graphs of Gol t al. [GKK10], to driv an fficint procdur for rounding fractional s-t flows and bipartit matchings. Part of this work was don whn th author was with Microsoft Rsarch Nw England. 1 W rcall that Õ(f) dnots O(f logc f), for som constant c.

2 1 Introduction Th maximum s-t flow problm and its dual, th minimum s-t cut problm, ar two of th most fundamntal and xtnsivly studid graph problms in combinatorial optimization [Sch03, AMO93, Sch02]. Thy hav a wid rang of applications (s [AMOR95]), ar oftn usd as subroutins in othr algorithms (s,.g., [AHK12, Sh09]), and a numbr of othr important problms.g., bipartit matching problm [CLRS09] can b rducd to thm. Furthrmor, ths two problms wr oftn a tstbd for dvlopmnt of fundamntal algorithmic tools and concpts. Most prominntly, th Max-Flow Min-Cut thorm [EFS56, FF56] constituts th prototypical primal-dual rlation. Svral dcads of xtnsiv work rsultd in a numbr of dvlopmnts on ths problms (s Goldbrg and Rao [GR98] for an ovrviw) and many of thir gnralizations and spcial cass. Still, dspit all this ffort, th basic problm of computing maximum s-t flow and minimum s-t cut in gnral graphs has rsistd progrss for a long tim. In particular, th currnt bst running tim bound of O(m min{m 1 2, n 2 3 } log(n 2 /m) log U) (with U dnoting th largst intgr arc capacity) was stablishd ovr 15 yars ago in a brakthrough papr by Goldbrg and Rao [GR98] and this bound, in turn, matchs th O(m min{m 1 2, n 2 3 }) bound for unit-capacity graphs that Evn and Tarjan [ET75] and, indpndntly, Karzanov [Kar73] put forth ovr 35 yars ago. Rcntly, howvr, important progrss was mad in th contxt of undirctd graphs. Christiano t al. [CKM + 11] dvlopd an algorithm that allows on to comput a (1 + ε)-approximation to th undirctd maximum s-t flow (and th minimum s-t cut) problm in Õ(mn 1 3 ε 11/3 ) tim. Thir rsult rlis on dvising a nw approach to th problm that combins lctrical flow computations with multiplicativ wights updat mthod (s [AHK12]). Latr, L t al. [LRS13] prsntd a quit diffrnt but still lctrical-flow-basd algorithm that mploys purly gradint-dscnttyp viw to obtain an Õ(mn1/3 ε 2/3 )-tim (1 + ε)-approximation for th cas of unit capacitis. Finally, vry rcntly, this lin of work was culminatd by Shrman [Sh13] and Klnr t al. [KLOS14] who indpndntly showd how to intgrat non-euclidan gradint-dscnt mthods with fast poly-logarithmic-approximation algorithms for cut problms of Mądry [Mąd10] to gt an O(m 1+o(1) ε 2 )-tim (1 + ε)-approximation to th undirctd maximum flow problm. Finally, w not that, in paralll to th abov work that is focusd on dsigning waklypolynomial algorithms for th maximum s-t flow and minimum s-t cut problms, thr is also a considrabl intrst in obtaining running tim bounds that ar strongly-polynomial, i.., that do not dpnd on th valus of arc capacitis. Th currnt bst such bound is O(mn) and it follows by combining th algorithms of King t al. [KRT94] and Orlin [Orl13]. Bipartit Matching Problm. Anothr problm that w will b intrstd in is th (maximumcardinality) bipartit matching problm a fundamntal assignmnt problm with numrous applications (s,.g., [AMO93, LP86]) and long history, with its roots in th works of Frobnius [Fro12, Fro17] and König [Kön15, Kön16, Kön23] from th arly 20th cntury (s [Sch05]). Alrady in 1931, König [Kön31] and Egrváry [Eg31] providd first constructiv charactrization of maximum matchings in bipartit graphs. This charactrization can b turnd into a polynomialtim algorithm. Thn, in 1973, Hopcroft and Karp [HK73] and, indpndntly, Karzanov [Kar73] dvisd th clbratd O(m n)-tim algorithm. Till dat, this bound is th bst on known in th rgim of rlativly spars graphs. It can b improvd, howvr, whn th input graph is dns, i.., whn m is clos to n 2. In this cas, on can combin th algbraic approach of Rabin 1

3 and Vazirani [RV89] that itslf builds on th work of Tutt [Tut47] and Lovász [Lov79] with matrix-invrsion tchniqus of Bunch and Hopcroft [BH74] to gt an algorithm that runs in O(n ω ) tim (s [Muc05]), whr ω is th xponnt of matrix multiplication [CW90, VW12]. Also, latr on, Alt t al. [ABMP91], as wll as, Fdr and Motwani [FM95] dvlopd combinatorial n algorithms that offr a slight improvmnt by a factor of, roughly, log 2 n m ovr th O(m n) bound of Hopcroft and Karp whnvr th graph is sufficintly dns. Finally, it is worth mntioning that thr was also a lot of dvlopmnts on th (maximumcardinality) matching problm in gnral, i.., not ncssarily bipartit, graphs. Starting with th pionring work of Edmonds [Edm65], ths dvlopmnts ld to bounds that ssntially match th running tim guarants that wr prviously known only for bipartit cas. Mor spcifically, th running tim bound of O(m n) for th gnral-graph cas was obtaind by Micali and Vazirani [MV80, Vaz94] (s also [GT91] and [GK04]). Whil, building on th algbraic charactrization of th problm du to Rabin and Vazirani [RV89], Mucha and Sankowski [MS04] and thn Harvy [Har09] gav O(n ω )-tim algorithms for gnral graphs. 1.1 Our Contribution In this papr, w dvlop a nw algorithm for solving maximum s-t flow and minimum s-t cut problms in dirctd graphs. Mor prcisly, w prov th following thorm. Thorm 1.1. Lt G = (V, E) b a dirctd graph with m arcs and unit capacitis. For any two vrtics s and t, on can comput an intgral maximum s-t flow and minimum s-t cut of G in Õ(m 10 7 ) tim. This improvs ovr th long-standing O(m min{ m, n 2/3 }) running tim bound du to Evn and Tarjan [ET75] and, in particular, finally braks th Ω(n 3 2 ) running tim barrir for spars dirctd graphs. Furthrmor, by applying a wll-known rduction (s [CLRS09]), our nw algorithm givs th first improvmnt on th spars-graph cas of th sminal O(m n)-tim algorithms of Hopcroft- Karp [HK73] and Karzanov [Kar73] for th maximum-cardinality bipartit matching problm. Thorm 1.2. Lt G = (V, E) b an undirctd bipartit graph with m dgs, on can solv th 10 maximum-cardinality bipartit matching problm in G in Õ(m 7 ) tim. This, again, braks th 40-yar-old running tim barrir of Ω(n 3 2 ) for this problm in spars graphs. Additionally, w dsign a simpl rduction of th maximum s-t flow problm to prfct bipartit b-matching problm (s Thorm 3.1). (This rduction can b sn as an adaptation of th rduction of th maximum vrtx-disjoint s-t-path problm to th bipartit matching problm du to Hoffman [Hof60] cf. Sction 16.7c in [Sch03]. 2 ) As th rduction in th othr dirction is wllknown alrady, this stablishs an algorithmic quivalnc of ths two problms. W also show (s Thorm 3.3 and Corollary 3.4) how this rduction, togthr with th sub-linar-tim algorithm for prfct matching problm in rgular bipartit graphs of Gol t al. [GKK10], lads to an fficint, narly-linar tim, rounding procdur for s-t flows. 3 2 W thank Lap Chi Lau [Lau13] for pointing out this similarity. 3 Rcntly, it cam to our attntion that a vry similar rounding rsult was indpndntly obtaind by Khanna t al. [KKL13]. 2

4 Finally, our main tchnical contribution is a primal-dual algorithm for (nar-)prfct bipartit b-matching problm (s Thorm 3.2). This itrativ algorithm draws on idas undrlying intriorpoint mthods and th lctrical flow framwork of Christiano t al. [CKM + 11]. It mploys lctrical flow computations to gradually improv th quality of maintaind solution by advancing it toward (nar-)optimality along so-calld cntral path. W dvlop a way of analyzing this algorithm s rat of convrgnc by rlating it to th structur of th corrsponding lctrical flows (s Thorm 5.5). This undrstanding nabls us to dvis a way of prturbing (s Sction 6.1) and prconditioning (s Sction 6.2) our intrmdiat solutions to nsur a convrgnc in only Õ(m 3 7 ) itrations and thus improv ovr th wll-known barrir of Ω(m 1 2 ) itrations that all th prvious intrior-point-mthods-basd algorithms suffr from. (To th bst of our knowldg, this is th first tim that this barrir was brokn for a natural optimization problm.) W also not that most of this undrstanding of convrgnc bhavior of intrior-point mthods can b carrid ovr to gnral LP stting. Thrfor, w ar hopful that our tchniqus can b xtndd and will vntually lad to braking th Ω(m 1 2 ) itrations barrir for gnral intrior-point mthods. 1.2 Our Approach Th cor of our approach compriss two componnts. On of thm is combinatorial in natur and xploits an intimat connction btwn th maximum s-t flow problm and bipartit matching problm. Th othr on is mor linar-algbraic and rlis on intrplay of intrior-point mthods and lctrical flows. Maximum flows and bipartit matchings. Th combinatorial componnt shows that not only on can rduc bipartit matching problm to th maximum s-t flow problm, but also that a rduction in th othr dirction xists. Namly, on can rduc, in a simpl and purly combinatorial way, th maximum s-t flow problm to a crtain variant of bipartit matching problm (s Thorm 3.1). Onc this rduction is stablishd, it allows us to shift our attntion to th matching problm. Also, as a byproduct, this rduction togthr with th algorithm of Gol t al. [GKK10] yilds a fast procdur for rounding fractional maximum flows (s Corollary 3.4). This nabls us to focus on obtaining solutions that ar only narly-optimal, instad of bing optimal. Bipartit Matchings and Elctrical Flows. Th othr componnt is basd on using th intrior-point mthod framwork in conjunction with narly-linar tim lctrical flow computations, to dvlop a fastr algorithm for th bipartit matching problm. Th point of start hr is a ralization that th rcnt approachs to approximating undirctd maximum flow [CKM + 11, LRS13, Sh13, KLOS13], dspit achiving imprssiv progrss, hav fundamntal limitations that mak thm unlikly to yild improvmnts for th xact undirctd or (approximat) dirctd stting. 4 Vry roughly spaking, ths limitations stm from th fact that, at thir cor, all ths algorithms mploy som vrsion of gradint-dscnt mthod that rlis on purly primal argumnts, whil almost compltly nglcting th dual aspct of th problm. It is wll-undrstood, howvr, that gtting a running tim guarant that dpnds logarithmically, 4 Not that it is known s,.g., [Mąd11] that computing xact maximum s-t flow in undirctd graphs is algorithmically quivalnt to computing th xact or approximat maximum s-t flow in dirctd graph. 3

5 instad of polynomially, on ε 1 and such dpndnc is a prrquisit to making progrss in dirctd stting on nds to also mbrac th dual sid of th problm and tak full advantag of it. Intrior-point mthods and fast algorithms. Th abov ralization motivats us to considr a mor sophisticatd approach, on that is inhrntly primal-dual and achivs logarithmic dpndnc on ε 1 : intrior-point mthods. Ths mthods constitut a powrful optimization paradigm that is a cornrston of convx optimization (s,.g., [BV04, Wri97, Y97]) and alrady ld to dvlopmnt of polynomial-tim xact algorithms for a varity of problms. Unfortunatly, dspit all its advantags and succsss in tackling hard optimization tasks, this paradigm has crtain shortcomings in th contxt of dsigning fast algorithms. Th main rason for that is th fact that ach itration of intrior-point mthod rquirs solving of a linar systm, a task for which th currnt fastst gnral-purpos algorithm runs in O(n ω ) = O(n ) tim [AHU74, CW90, VW12]. So, this bound bcoms a bottlnck if on was aiming for, say, vn sub-quadratic-tim algorithm. Fortunatly, it turns out that thr is a way to circumvnt this issu. Namly, vn though th abov bound is th bst on known in gnral, on can gt a bttr running tim whn daling with som spcific problm. This is achivd by xploiting th spcial structur of th corrsponding linar systms. A prominnt (and most important from our point of viw) xampl hr is th family of flow problms. Daitch and Spilman [DS08] showd that in th contxt of flow problms on can us th powr of fast (approximat) Laplacian systm solvrs [?, KMP10, KMP11, KOSZ13] to solv th corrsponding linar systms in narly-linar tim. This nabld [DS08] to dvlop a host of Õ(m 3 2 )-tim algorithms for a numbr of important gnralizations of th maximum flow problm for which thr was no such algorithms bfor. Unfortunatly, this bound of Õ(m 3 2 ) tim turns out to also b a barrir if on wants to obtain vn fastr algorithms. Th nw difficulty hr is that th bst worst-cas bound on th numbr of itrations ndd for an intrior-point mthod to convrg to nar-optimal solution is Ω(m 1/2 ). Although it is widly blivd that this bound is far from optimal, it sms that our thortical undrstanding of intrior-point mthod convrgnc is still insufficint to mak any progrss on this front. In fact, improving this stat of affairs is a major and long-standing challng in mathmatical programing. Byond th Ω(m 1 2 ) barrir. Our approach to circumvnting this Ω(m 1 2 ) barrir and obtaining 10 th dsird Õ(m 7 )-tim algorithm for th bipartit b-matching problm consists of two stags. First on prsntd in Sction 5 corrsponds to stting up a primal-dual framwork for solving th nar-prfct b-matching problm. This framwork is dirctly inspird by th principls undrlying path-following intrior-point mthods and, in som sns, is quivalnt to thm. In it, w start with som initial sub-optimal solution (that is ncodd as a minimum-cost flow problm instanc) and gradually improv its quality up to nar-optimality. Ths gradual improvmnts ar guidd by crtain lctrical flow computations th flows ar usd to updat th primal solution and th corrsponding voltags updat th dual on and our solution nds up following a spcial trajctory in th fasibl spac: so-calld cntral path. W analyz th prformanc of this optimization procss by stablishing a formal connction that tis th siz of ach improvmnt stp to a crtain charactristic of th corrsponding lctrical flow. Vry roughly spaking, this siz (and thus th rsulting rat of convrgnc) is dirctly rlatd to how much th lctrical flow w comput rsmbls th currnt primal solution (which is also a 4

6 flow). Onc this connction is stablishd, a simpl nrgy-basd argumnt immdiatly rcovrs th gnric O(m 1 2 ) itrations bound known for intrior-point mthods. So, as ach lctrical flow computation can b prformd in Õ(m) tim, this givs an ovrall Õ(m 3 2 )-tim algorithm. Finally, to improv upon th abov O(m 1 2 ) itrations bound and dlivr th dsird O(m 10 7 )- tim procdur, in Sction 6, w dvis two tchniqus: prturbation of arcs that can b sn as a rfinmnt of th dg rmoval tchniqu of Christiano t al. [CKM + 11]; and solution prconditioning a way of adding auxiliary arcs to th solution to improv its conductanc proprtis. W show that by a carful composition of ths tchniqus, on is abl to nsur that th guiding lctrical flows align bttr with th primal solution thus allowing taking largr progrss stps and guaranting fastr convrgnc whil kping th unwantd impact of ths modifications on th quality of final solution minimal. Th analysis of this procss constituts th tchnical cor of our rsult and is basd on undrstanding of th intrplay btwn th intrior-point mthod and both th primal and dual structur of lctrical flows. W bliv that this approach of undrstanding intrior-point mthods through th lns of lctrical flows is a promising dirction and our rsult is just a first stp towards ralizing its full potntial. 1.3 Organization W bgin th tchnical part of th papr in Sction 2 whr w prsnt som prliminaris on maximum flow problm, lctrical flows, and bipartit (b-)matching problm, as wll as, introduc som thorms w will nd in th squl. In Sction 3, w provid a gnral outlin of our rsults and th structur of our proof. In Sction 4, w dscrib th rduction of maximum s-t flow problm to th bipartit b-matching problm. Nxt, in Sctions 5 and 6, w xplain how our path-following algorithms and lctrical flows can b usd to gt an improvd algorithm for th bipartit b-matching problm, with Sction 7 prsnting th analysis of our path-following primitiv. Finally, w conclud in Sction 8 by showing how to round fractional b-matchings to intgral ons. 2 Prliminaris In this sction, w introduc som basic notation and dfinitions w will nd latr. 2.1 σ-flows and th Maximum s-t Flow Problm Throughout this papr, w dnot by G = (V, E, u) a dirctd graph with vrtx st V, arc st E (w allow paralll arcs), and (non-ngativ) intgr capacitis u, for ach arc E. W usually dfin m = E to b th numbr of arcs of th graph in qustion and n = V to b th numbr of its vrtics. Each arc of G is an ordrd pair (u, v), whr u is its tail and v is its had. Th basic notion of this papr is th notion of a σ-flow in G, whr σ R n, with v σ v = 0, is th dmand vctor. By a σ-flow in G w undrstand any vctor f R m that assigns valus to arcs G and satisfis th flow consrvation constraints: f f = σ v, for ach vrtx v V. (1) E + (v) E (v) 5

7 Hr, E + (v) (rsp. E (v)) is th st of arcs of G that ar laving (rsp. ntring) vrtx v. Intuitivly, ths constraints nforc that th nt balanc of th total in-flow into vrtx v and th total out-flow out of that vrtx is qual to σ v, for vry v V. Furthrmor, w say that a σ-flow f is fasibl in G iff f obys th non-ngativity and capacity constraints: 0 f u, for ach arc E. (2) On typ of σ-flows that will b of spcial intrst to us ar s-t flows, whr s (th sourc) and t (th sink) ar two distinguish vrtics of G. Formally, a σ-flow f is an s-t flow iff its dmand vctor σ is qual to F χ s,t for som F 0 w call F th valu of f and th dmand vctor χ s,t that has 1 (rsp. 1) at th coordinat corrsponding to s (rsp. t) and zros vrywhr ls. Now, th maximum s-t flow problm corrsponds to a task of finding for a givn graph G = (V, E, u), a sourc s, and a sink t, a fasibl s-t flow f in G of maximum valu F. W call such a flow f that maximizs F th maximum s-t flow of G and dnot its valu by F. Somtims, w will b also intrstd in (uncapacitatd) minimum-cost σ-flow problm (with non-ngativ costs). In this problm, w hav a dirctd graph G with infinit capacitis on arcs (i.., u = +, for all ) and crtain (non-ngativ) lngth (or cost) l assignd to ach arc. Our goal is to find a fasibl σ-flow f in G whos cost l(f ) := l f is minimal. (Not that as w hav infinit capacitis hr, th fasibility constraint (2) just rquirs that f 0 for all arcs.) Finally, on mor problm that will b rlvant in this contxt is th minimum s-t cut problm. In this problm, w ar givn a dirctd graph G = (V, E, u) with intgr capacitis, as wll as, a sourc s and sink t, and our task is to find an s-t cut C V in G minimizs th capacity u(c) := E (C) u among all s-t cuts. Hr, a cut C V is an s-t cut iff s C and t / C, and E (C) is th st of all arcs (u, v) with u C and v / C. It is wll-known [EFS56, FF56] that th minimum s-t cut problm is th dual of th maximum s-t problm and, in particular, that th capacity of th minimum s-t cut is qual to th valu of th maximum s-t flow, as wll as, that givn a maximum s-t flow on can asily obtain th corrsponding minimum s-t cut. 2.2 Undirctd Graphs Although th focus of our rsults is on dirctd graphs, it will b crucial for us to considr undirctd graphs too. To this nd, w viw an undirctd graph G = (V, E, u) as a dirctd on in which th ordrd pair (u, v) E dos not dnot an arc anymor, but an (undirctd) dg (u, v) and th ordr just spcifis an orintation of that dg from u to v. (Evn though w us th sam notation for ths two diffrnt typs of graphs, w will always mak sur that it is clar from th contxt whthr w dal with dirctd graph that has arcs, or with undirctd graph that has dgs.) From this prspctiv, th dfinitions of σ-flow f that w introducd abov for dirctd graphs transfr ovr to undirctd stting almost immdiatly. Th only (but vry crucial) diffrnc is that in undirctd graphs a fasibl flow can hav som of f s bing ngativ - this corrsponds to th flow flowing in th dirction that is opposit to th dg orintation. As a rsult, th fasibility condition (2) bcoms f u, for ach arc E. (3) Also, th st E + (v) (rsp. E (v)) dnots now th st of incidnt dgs that ar orintd towards (rsp. away) from v, and E(v) := E + (v) E (v) is just th st of all dgs incidnt to v, rgardlss of thir orintation. 6

8 Finally, givn a dirctd graph G = (V, E, u), by its projction Ḡ w undrstand an undirctd graph that ariss from trating ach arc of G as an dg with th corrsponding orintation. Not that if G had two arcs (u, v) and (v, u) thn Ḡ will hav two paralll dgs (u, v) and (v, u) that hav opposit orintation and, possibly, diffrnt capacitis. 2.3 Elctrical Flows and Potntials A notion that will play a fundamntal rol in this papr is th notion of lctrical flows. Hr, w just brifly rviw som of th ky proprtis that w will nd latr. For an in-dpth tratmnt w rfr th radr to [Bol98]. Considr an undirctd graph G and som vctor of rsistancs r R m that assigns to ach dg its rsistanc r > 0. For a givn σ-flow f in G, lt us dfin its nrgy (with rspct to r) E r (f ) to b E r (f ) := r f 2 = f T Rf, (4) whr R is an m m diagonal matrix with R, = r, for ach dg. For a givn undirctd graph G, a dmand vctor σ, and a vctor of rsistancs r, w dfin an lctrical σ-flow in G (that is dtrmind by rsistancs r) to b th σ-flow that minimizs th nrgy E r (f ) among all σ-flows in G. As nrgy is a strictly convx function, on can asily s that such a flow is uniqu. Also, w mphasiz that w do not rquir hr that this flow is fasibl with rspct to capacitis of G (cf. (3)). Furthrmor, whnvr w considr lctrical flows in th contxt of a dirctd graph G, w will man an lctrical flow as dfind abov in th (undirctd) projction Ḡ of G. On of vry usful proprtis of lctrical flows is that it can b charactrizd in trms of vrtx potntials inducing it. Namly, on can show that a σ-flow f in G is an lctrical σ-flow dtrmind by rsistancs r iff thr xist vrtx potntials φ v (that w collct into a vctor φ R n ) such that, for any dg = (u, v) in G that is orintd from u to v, f = φ v φ u r. (5) In othr words, a σ-flow f is an lctrical σ-flow iff it is inducd via (5) by som vrtx potntial φ. (Not that orintation of dgs mattrs in this dfinition.) Using vrtx potntials, w ar abl to xprss th nrgy E r (f ) (s (4)) of an lctrical σ-flow f in trms of th potntials φ inducing it as E r (f ) = =(u,v) (φ v φ u ) 2 r. (6) On of th consquncs of this charactrization of lctrical flows via vrtx potntials is that on can viw th nrgy of an lctrical σ-flow as bing a rsult of optimization not ovr all th σ-flows but rathr ovr crtain st of vrtx potntials. Namly, w hav th following lmma that, for compltnss, w prov in th Appndix A. Lmma 2.1. For any graph G = (V, E), any vctor of rsistancs r, and any dmand vctor σ, 1 E r (f ) = min φ σ T φ=1 =(u,v) E 7 (φ v φ u ) 2 r,

9 whr f is th lctrical σ-flow dtrmind by r in G. Furthrmor, if φ ar th vrtx potntials corrsponding to f thn th minimum is attaind by taking φ to b qual to φ := φ /E r (f ). Not that th abov lmma provids a convnint way of lowrbounding th nrgy of an lctrical σ-flow. On just nds to xpos any vrtx potntials φ such that σ T φ = 1 and this will immdiatly constitut an nrgy lowrbound. Also, anothr basic but usful proprty of lctrical σ-flows is capturd by th following fact. Fact 2.2 (Rayligh Monotonicity). For any graph G = (V, E), dmand vctor σ and any two vctors of rsistancs r and r such that r r, for all E, w hav that if f (rsp. f ) is th lctrical σ-flow dtrmind by r (rsp. r ) thn 2.4 Laplacian Solvrs E r (f ) E r (f ). A vry important algorithmic proprty of lctrical flows is that on can comput vry good approximations of thm in narly-linar tim. Blow, w brifly dscrib th tools nabling that. To this nd, lt us rcall that lctrical σ-flow is th (uniqu) σ-flow inducd by vrtx potntials via (5). So, finding such a flow boils down to computing th corrsponding vrtx potntials φ. It turns out that computing ths potntials can b cast as a task of solving crtain typ of linar systm calld Laplacian systms. To s that, lt us dfin th dg-vrtx incidnc matrix B bing an n m matrix with rows indxd by vrtics and columns indxd by dgs such that 1 if E + (v), B v, = 1 if E (v), 0 othrwis. Now, w can compactly xprss th flow consrvation constraints (1) of a σ-flow f (that w viw as a vctor in R m ) as Bf = σ. On th othr hand, if φ ar som vrtx potntials, th corrsponding flow f inducd by φ via (5) (with rspct to rsistancs r) can b writtn as f = R 1 B T φ, whr again R is a diagonal m m matrix with R, := r, for ach dg. Putting th two abov quations togthr, w gt that th vrtx potntials φ that induc th lctrical σ-flow dtrmind by rsistancs r ar givn by a solution to th following linar systm BR 1 B T φ = Lφ = σ, (7) whr L := BR 1 B T is th (wightd) Laplacian L of G (with rspct to th rsistancs r). On can asily chck that L is an n n matrix indxd by vrtics of G with ntris givn by E(v) 1/r if u = v, L u,v = 1/r if = (u, v) E, and (8) 0 othrwis. 8

10 On can s that th Laplacian L is not invrtibl, but as long as, th undrlying graph is connctd it s null-spac is on-dimnsional and spannd by all-ons vctor. As w rquir our dmand vctors σ to hav its ntris sum up to zro (othrwis, no σ-flow can xist), this mans that thy ar always orthogonal to that null-spac. Thrfor, th linar systm (7) has always a solution φ and on of ths solutions 5 is givn by φ = L σ, whr L is th Moor-Pnros psudo-invrs of L. Now, from th algorithmic point of viw, th crucial proprty of th Laplacian L is that it is symmtric and diagonally dominant, i.., for any v V, u v L u,v L v,v. This nabls us to us fast approximat solvrs for symmtric and diagonally dominant linar systms to comput an approximat lctrical σ-flow. Namly, building on th work of Spilman and Tng [ST03, ST04], Koutis t al. [KMP10, KMP11] dsignd an SDD linar systm solvr that implis th following thorm. (S also rcnt work of Klnr t al. [KOSZ13] that prsnts an vn simplr narlylinar-tim Laplacian solvr.) Thorm 2.3. For any ε > 0, any graph G with n vrtics and m dgs, any dmand vctor σ, and any rsistancs r, on can comput in Õ(m log m log ε 1 ) tim vrtx potntials φ such that φ φ L ε φ L, whr L is th Laplacian of G, φ ar potntials inducing th lctrical σ-flow dtrmind by rsistancs r, and φ L := φ T Lφ. To undrstand th typ of approximation offrd by th abov thorm, obsrv that φ 2 L = φ T Lφ is just th nrgy of th flow inducd by vrtx potntials φ. Thrfor, φ φ L is th nrgy of th lctrical flow f that corrcts th vrtx dmands of th lctrical σ-flow inducd by potntials φ, to th ons that ar dictatd by σ. So, in othr words, th abov thorm tlls us that w can quickly find an lctrical σ-flow f in G such that σ is a slightly prturbd vrsion of σ and f can b corrctd to th lctrical σ-flow f that w ar sking, by adding to it som lctrical flow f whos nrgy is at most ε fraction of th nrgy of th flow f. (Not that lctrical flows ar linar, so w indd hav that f = f + f.) As w will s, this kind of approximation is compltly sufficint for our purposs. 2.5 Bipartit b-matchings A fundamntal graph problm that constituts both an application of our rsults, as wll as, on of th tools w us to stablish thm, is th (maximum-cardinality) bipartit b-matching problm. In this problm, w ar givn an undirctd bipartit graph G = (V, E) with V = P Q whr P and Q ar th two sts of bipartition as wll as, a dmand vctor b that assigns to vry vrtx v an intgral and positiv dmand b v. Our goal is to find a maximum cardinality multist M of th dgs of G that forms a b-matching. That is, w want to find a multi-st M of dgs of G that is of maximum cardinality subjct to a constraint that, for ach vrtx v V, th numbr of dgs of M that ar incidnt to v is at most b v. (Whn b v = 1 for vry vrtx v, w will simply call such M a matching.) W say that a b-matching M is prfct iff vry vrtx in V has xactly b v dgs incidnt to it in M. Not that a prfct b-matching - if it xists in G - has to ncssarily b of maximum 5 Not that th linar systm (7) will hav many solutions, but ach two of thm ar quivalnt up to a translation. So, as th formula (5) is translation-invariant, ach of ths solutions will yild th sam uniqu lctrical σ-flow. 9

11 cardinality. Also, if a graph has a prfct b-matching thn it must b that v P b v = v Q b v. Now, by th prfct bipartit b-matching problm w man a task in which w nd to ithr find th prfct b-matching in G or conclud that it dos not xist. Finally, by a fractional solution to a b-matching problm, w undrstand an E -dimnsional vctor x that allocats non-ngativ valu of x to ach dg and is such that for vry vrtx v of G, th sum E(v) x of (fractional) incidnt dgs in x is at most b v. Also, w dfin th siz of a fractional b-matching x to b x 1. An intrsting class of graphs that is guarantd to always hav a prfct matching ar bipartit graphs that ar d-rgular, i.., that hav th dgr of ach vrtx qual to d. A rmarkabl algorithm of Gol t al. [GKK10] shows that on can find a prfct matching in such graphs in tim that is proportional only to numbr of its vrtics and not dgs. (Not that a d-rgular bipartit graph has xactly dn 2 dgs and thus this numbr can b much highr than n whn d is larg.) In particular, thy prov th following thorm that w will us latr. Thorm 2.4 (s Thorm 4 in [GKK10]). Givn an n n doubly-stochastic matrix M with m non-zro ntris, on can find a prfct matching in th support of M in O(n log 2 n) xpctd tim with O(m) prprocssing tim. 3 From Flows to Matchings, and Back As w alrady mntiond, our rsults stm from xploiting th intrplay btwn th maximum s-t flow and bipartit b-matching problm, as wll as, from undrstanding th prformanc of intriorpoint mthods whn applid to ths two problms via th structur of corrsponding lctrical flows. To highlight ths lmnts, w dcompos th proof of our main thorm (Thorm 1.1) into thr natural parts. Rducing Maximum Flow to b-matching First, w focus on analyzing th rlationship btwn th maximum s-t flow and th (maximumcardinality) bipartit b-matching problm. It is wll-known that th lattr can b rducd to th formr in a simpl way. As it turns out, howvr, on can also go th othr way thr is a simpl, combinatorial rduction from th maximum flow problm to th task of finding a prfct bipartit b-matching. 6 Bfor making this prcis, lt us introduc on dfinition. Considr a b-matching problm instanc corrsponding to a bipartit graph G = (V, E) with P and Q (V = P Q) bing two sids of th bipartition. For any dg = (p, q) E, lt us dfin th thicknss d() of that dg to b d() := min{b p, b q }. (So, d() is an uppr bound on th valu of x in any fasibl b-matching x.) W say that a b-matching instanc is balancd iff d() 4 b 1. (9) E Now, in Sction 4, w stablish th following rsult. 6 On can viw this as on possibl xplanation of why th tchniqus usd in th contxt of bipartit matchings and maximum flows ar so similar. 10

12 Thorm 3.1. If on can solv a balancd instanc of a prfct bipartit b-matching problm in a (bipartit) graph with n vrtics and m dgs in T ( n, m, b 1 ) tim, thn on can solv th maximum s-t flow problm in a graph G = (V, E, u) with m arcs and capacity vctor u in Õ((m + T (Θ(m), 4m, 4 u 1 )) log u 1 ) tim. This connction btwn maximum flows and bipartit matchings is usful in two ways. Firstly, it nabls us to rduc th main problm w want to solv th maximum s-t flow problm with unit capacitis to a smingly simplr on: th prfct bipartit b-matching problm. Scondly, th fact that this rduction works also for fractional instancs provids us with an ability to lift our b-matching rounding procdur that w dvlop latr (s Thorm 3.3) to th maximum flow stting (s Corollary 3.4). Th Algorithm for Nar-Prfct b-matching Problm Onc th abov rduction is stablishd, w can procd to dsigning an improvd algorithm for th prfct bipartit b-matching problm. This algorithm consists of two parts. Th first on constituting th tchnical cor of our papr is rlatd to th (fractional) nar-prfct bipartit b-matching problm, a crtain rlaxation of th prfct bipartit b-matching problm. To dscrib this task formally, lt us call a b-matching x nar-prfct if its siz x 1 is at last b 1 2 Õ(m 3 7 ), i.., it is within Õ(m 3 7 ) additiv factor of th siz of a prfct b-matching. Now, givn a bipartit graph G = (P Q, E) and dmand vctor b, th nar-prfct b-matching problm is a task of ithr finding a nar-prfct b-matching in G or concluding that no prfct b-matching xists in that graph. Our goal is to dsign an algorithm that solvs this nar-prfct b-matching problm in Õ(m 10 7 ) tim. To this nd, in Sctions 5 and 6 w prov th following thorm. Thorm 3.2. Lt G = (V, E) with V = P Q b an undirctd bipartit graph with n vrtics and m dgs and lt b b a dmand vctor that corrsponds to a balancd b-matching instanc with b 1 = O(m). In Õ(m 10 7 ) tim, on can ithr find a fractional nar-prfct b-matching x or conclud that no prfct b-matching xists in G. (Obsrv that whnvr w hav an instanc of maximum s-t flow problm that has m arcs and unit capacitis, u 1 is xactly m. So, if w apply th rduction from Thorm 3.1 to that instanc thn th rsulting b-matching problm instanc will b balancd, hav m 4 m dgs, as wll as, b 1 4 u 1 = 4 m 2m. Thrfor, w will b abl to apply th abov Thorm 3.2 to it.) At a vry high lvl, our algorithm for th nar-prfct b-matching problm is inspird by th way th xisting intrior-point mthod path-following algorithms (s,.g., [Y97, Wri97, BV04]) can b usd to solv it. Basically, our algorithm is an itrativ mthod that starts with som initial, far-from-optimal solution and thn gradually improvs this maintaind solution to nar-optimality (pushing it along so-calld cntral path) using appropriat lctrical flows as a guidanc. W thn show how to ti th convrgnc rat of this procss to th structur of th guiding lctrical flows. At that point, on can us a simpl nrgy-bounding argumnt to stablish a gnric convrgnc bound that yilds an (unsatisfactory) Õ(m 3 2 )-tim algorithm. To improv upon this bound and dlivr th dsird Õ(m 10 7 )-tim algorithm, w show how on can appropriatly shap ths guiding lctrical flows to mak thir guidanc mor ffctiv and thus guarant fastr convrgnc. Vry roughly spaking, it turns out thr is a way of changing 11

13 th maintaind solution to mak it ssntially th sam from th point of viw of our b-matching instanc, whil dramatically improving th quality of corrsponding lctrical flows that guid it. Our way of xcuting this ida is basd on a carful composition of two tchniqus. On of thm corrsponds to prturbing, in a crtain way, th arcs that ar most significantly distorting th structur of lctrical flow this tchniqu can b viwd as a rfinmnt of dg rmoval tchniqu of Christiano t al. [CKM + 11]. Th othr tchniqu corrsponds to prconditioning th whol solution by adding additional, auxiliary, arcs to it. Ths arcs ar chosn so to significantly improv th conductanc proprtis of th solution (whn viwd as a graph with rsistancs) whil not lading to too significant dformation of th final obtaind solution. Rounding Nar-Prfct b-matchings Finally, our final stp on our way towards solving th prfct b-matching problm (and thus th maximum s-t flow problm) is rlatd to turning th approximat and fractional answr rturnd by th algorithm from Thorm 3.2 into an xact and intgral on. To this nd, not that if that algorithm rturnd a nar-prfct b-matching that was intgral, thr would b a standard way to ithr turn it into a prfct b-matching or conclud that no such prfct b-matching xists. Namly, on could just us rpatd augmnting path computations. It is wll-known that givn an intgral b-matching, on can prform, in O(m) tim, an augmnting path computation that ithr rsults in incrasing th siz of our b-matching by on, or concluds that no furthr augmntation is possibl (and thus no prfct b-matching xists). So, as our initial nar-prfct b-matching has siz at last b 1 2 Õ(m 3 7 ), aftr at most Õ(m 3 7 ) itrations, i.., in tim Õ(m 10 7 ), w would gt th dsird answr. Unfortunatly, th abov approach can fail compltly onc our nar-prfct b-matching is fractional. This is so, as in this cas w do not hav any maningful lowrbound on th progrss on th siz of th b-matching brought by th augmnting path computation. Thrfor, to dal with this issu, w dvlop th last ingrdint of our algorithm: a narlylinar tim procdur that allows on to round fractional b-matchings. Mor prcisly, in Sction 8, building on th work of Gol t al. [GKK10] (s Thorm 2.4), w stablish th following thorm. Thorm 3.3. Lt G = (V, E) b an undirctd bipartit graph with m dgs and lt b b a dmand vctor, if x is a fractional b-matching in G of siz k = x 1 thn on can find in Õ(m) tim an intgral b-matching in G of siz k. Clarly, if w apply th abov rounding mthod to th fractional nar-prfct matching x computd by th algorithm from Thorm 3.2, it will giv us an intgral b-matching x whos siz is still at last b 1 2 Õ(m 3 7 ). So, th augmnting path-basd approach w outlind abov will lt us obtain th dsird intgral and xact answr to th prfct b-matching problm within th dsird tim bound. In th light of all th abov, w s that combining all th abov pics indd yilds an Õ(m 10 7 )- tim algorithm for th prfct bipartit b-matching problm in graphs with b 1 = O(m). Now, using th rduction from Thorm 3.1, this givs us th analogous algorithm for th maximum s-t flow problm in unit-capacity graphs and that, in turn, rsults in an algorithm for th bipartit matching problm. So, both Thorm 1.1 and Thorm 1.2 hold. 12

14 Rounding s-t Flows Finally, w mntion th othr byproduct of our tchniqus th fast rounding procdur for flows. Namly, using th rduction dscribd in Thorm 3.1 and th rounding from Thorm 3.3 w can obtain a fast rounding procdur not only for fractional b-matchings but also for fractional s-t flows. Spcifically, th proof of th following corollary appars in Appndix B. Corollary 3.4. Lt G = (V, E, u) b a dirctd graph with capacitis and lt f b som fasibl fractional s-t flow in G of valu F. In Õ(m) tim, w can obtain out of f an intgral s-t flow f of valu F that is fasibl in G. Again, w not that a vry similar rounding rsult was indpndntly obtaind by Khanna t al. [KKL13]. 4 From Maximum Flows to Prfct Matchings In this sction, w show how to rduc th maximum s-t flow problm in a dirctd capacitatd graph G = (V, E, u) to solving O(log u 1 ) balancd instancs of th prfct bipartit b-matching problm, i.., w prov Thorm 3.1. W not that our rduction can b sn as an adaptation of th rduction of th maximum vrtx-disjoint s-t path problm to th bipartit matching problm du to Hoffman [Hof60] cf. Sction 16.7c in [Sch03]. To this nd, lt G = (V, E, u) with n = V vrtics and m = E arcs, as wll as, th sourc s and sink t b our input instanc of th maximum s-t flow problm. Without loss of gnrality, w can assum that thr is no arcs ntring s and no arcs laving t, as ths arcs do not affct th maximum s-t flow. Also, lt F b th valu of th maximum s-t flow in G. 4.1 Th Rduction W show that for any intgral valu of F, w can stup, in Õ(m) tim, a balancd bipartit b- matching problm instanc, for som dmands b and bipartit graph Ḡ = (P Q, Ē), such that: (1) thr will b a prfct b-matching in Ḡ if thr is a fasibl s-t flow of valu F in G; and (2) givn a prfct b-matching in Ḡ on can rcovr in Õ(m) tim an s-t flow of valu F that is fasibl in G. Obsrv that onc such a rduction is dsignd, Thorm 3.1 will follow by noticing that 1 F u 1 and applying a simpl binary sarch stratgy to find th valu of F and xtract th corrsponding maximum s-t-flow. Givn th input graph G = (V, E, u), sourc s, sink t and th valu of F, th construction of our dsird balancd bipartit b-matching instanc Ḡ = (P Q, Ē) is as follows. First, for ach arc E, w crat two vrtics p P and q Q and an dg (p, q ) btwn thm, as wll as, w st th dmand b p and b q of ths vrtics to u. Nxt, for vry vrtx v of G othr than s and t, w add a vrtx p v to P and a vrtx q v to Q. Also, w crat an dg (p v, q v ), as wll as, an dg (p v, q ) (rsp. (q v, p )) for vry arc that is incoming to (rsp. outgoing of) v in G. W st th dmands b pv (rsp. b qv ) to b qual to E + (v) u (rsp. E (v) u ). Finally, w crat a vrtx q s Q (rsp. p t P ) and add an dg (q s, p ) (rsp. (q, p t ) for ach arc that is laving s (rsp. incoming to t) in G. W put th dmand b qs (rsp. b pt ) to b ( E (s) u ) F (rsp. ( E + (t) u ) F ). (Not that w can assum hr that both ths quantitis ar non-ngativ as both E (s) u and E + (t) u ar obvious upprbounds on th valu of F.) 13

15 a) t b) v s v p 1 t q 3 3 q 4 1 p 4 3 p 3 3 q v q v2 p v1 p 5 q 5 p v q 1 p q s p 2 q 2 Figur 1: a) An xampl dirctd s-t flow instanc G. Numbrs nxt to arcs dnot thir capacitis. b) Th b-matching instanc corrsponding to th xampl from a) in cas of F = 2. Hr, numbrs nxt to vrtics dnot thir dmands. An xampl s-t flow instanc and th corrsponding instanc of th bipartit b-matching can b found in Figur 1. To s that this instanc is balancd, not that vry dg h of Ḡ that is incidnt to som vrtx p or q has its thicknss d(h) qual to u = b p = b q. So, th contribution of ths dgs to th total thicknss h Ē d(h) of dgs of Ḡ is at most 3 E u 3 2 b 1. On th othr hand, th only dgs that ar not incidnt to som p or q ar th ons of th form (p v, q v ). Howvr, th total contribution of ths dgs to th total thicknss is at most min{ u, u } E + (v) u + E (v) u u 1 b 1, 2 v s,t E + (v) E (v) v s,t as ndd. Now, th proof of corrctnss of this rduction appars in Appndix C. 5 Basic Õ(m 3 2)-Tim Algorithm for Bipartit b-matching Problm Ovr th nxt two sctions, w prov Thorm 3.2. That is, w prsnt an algorithm for th narprfct bipartit b-matching problm in th stting whr th input instanc is balancd (s (9)) and b 1 is O(m). In what follows w assum, for convninc, that b 1 is at most 2m and that th graph G is spars, i.., m = O(n). 7 7 It is asy to s that ths assumptions ar mad without loss of gnrality. Whnvr b 1 is O(m), on can nsur that b 1 2m and m = O(n) by adding an appropriat but still O(m) numbr of dummy copis of complt bipartit K 6,6 graph with uniform dmands. Adding ach such dummy isolatd copy brings th ratio of b 1 and m, as wll as, of m to n down towards 18, whil nvr lading to violation of th balanc condition (9) and 12 prsrving th b-matching structur of th original input graph. 14

16 In this sction, w show a basic algorithm that runs in Õ(m 3 2 ) tim. Latr, in Sction 6, w rfin this algorithm to obtain th dsird running tim of Õ(m 10 7 ). For th sak of clarity, in our dscription and analysis w assum that th narly-linar tim Laplacian systm solvr (s Thorm 2.3) always rturns an xact solution, i.., all th lctrical σ-flows w comput ar xact. W discuss how to handl th approximat natur of th solvr s output in Appndix E.9. From b-matching to Min-Cost σ-flow Lt us fix our instanc of th bipartit b-matching problm in bipartit graph G = (V, E) with V = P Q. W will solv our b-matching instanc by rducing it to a task of finding a minimumcost ˆσ-flow in a crtain rlatd dirctd graph Ĝ = ( V, Ê,ˆl) with ˆl bing a lngth vctor. a) b) q 1 q 2 q t q1 t q2 t q v p 1 p 2 p 3 p 4 s -1 p1 s -3 p2 s -1 p3 s p4 Figur 2: a) An xampl instanc of bipartit b-matching problm. Numbrs nxt to vrtics rprsnt thir dmands. b) Th minimum-cost ˆσ-flow problm instanc corrsponding to th xampl from a). All arcs hav cost ˆl qual to 1 and th numbrs nxt to vrtics dnot thir dmands in ˆσ. Thr ar two paralll copis of th arc (s p1, t q1 ) and thr paralll copis of th arc (s p3, t q3 ). Also, ach dashd arc rprsnts two arcs that hav th sam ndpoints but opposit orintation. Th rduction is prformd as follows (s Figur 2 for an xampl). Th vrtx st V of th graph Ĝ consist of a spcial vrtx v, as wll as, vrtics s p (rsp. t q ), for vry vrtx p P (rsp. q Q) of th graph G. Nxt, for vry dg = (p, q) in G, w add to Ĝ d() copis of an arc (s p, t q ), whr w rcall that d() := min{b p, b q } is th thicknss of. Finally, for ach vrtx p P (rsp. q Q) of G, w add to Ĝ arcs (s p, v ) and (v, s p ) (rsp. (v, t q ) and (t q, v )). W st th lngths ˆl of all arcs to 1. To gain som intuition on this rduction, not that if a prfct b-matching indd xists in G thn th flow that ncods it in Ĝ is fully supportd on th arcs (s p, t q ) and dos not snd mor than on unit of flow on any of ths arcs. So, th purpos of including th xtra vrtx v and th arcs incidnt to it is to support (and appropriatly pnaliz) th initial and intrmdiat solutions as thy approach optimality. Also, obsrv that this nw graph has ˆ:= n + 1 vrtics and, du to our b-matching instanc 15

17 bing balancd, w hav that th total numbr m of arcs is at most 2n + d() 2n + O(m) = O(m). =(p,q) G So, bounding our running tim in trms of m provids a bound in trms of th numbr of dgs m of our original b-matching instanc that is asymptotically th sam. Now, considr a dmand vctor ˆσ that has surplus of b p at ach vrtx s p, a dficit of b q at ach vrtx t q and a zro dmand at vrtx v. (Not that such a dmand vctor will b valid, i.., v ˆσ v = 0, as w can assum that p b p = q b q othrwis it would b impossibl to hav a prfct b-matching in G.) W claim that any nar-optimal ˆσ-flow givs us a solution to our nar-prfct b-matching instanc. (Rcall from Sction 3 that a b-matching is nar-prfct if its siz is at last b 1 2 Õ( m 3 7 ). Although, in th lmma blow it suffics that w hav a slack of only 1 2 instad of Õ( m 3 7 ).) Lmma 5.1. Givn any fasibl ˆσ-flow f in Ĝ whos cost ˆl(f ) is within additiv 1 2 of th optimum, in Õ( m) tim, w can ithr comput a (fractional) nar-prfct b-matching x in G or conclud that no prfct b-matching xists in G. Proof. First, obsrv that if thr xists a prfct b-matching x in G thn a flow f that just puts, x for ach = (p, q) of G, d() 1 units of flow on ach (of d()) copis of th arc (s p, t q ) in Ĝ, is a fasibl ˆσ-flow with cost b 1 2. (Rcall that in th minimum-cost problm w assum that arc capacitis ar infinit, thus fasibility condition (2) boils down to non-ngativity of all f s.) So, w can assum that our ˆσ-flow f has its cost ˆl(f ) at most b (Othrwis, w know that thr is no prfct b-matching in G.) Now, givn any fasibl ˆσ-flow in Ĝ, w can dcompos it into a collction of flow-paths and flow-cycls, whr ach of ths flow-paths transports som amount of flow from som vrtx s p to som vrtx t q. By our construction of th graph Ĝ, ach such flow-path has to hav a lngth at last 1. On th othr hand, if this flow-path is indd of lngth xactly 1 thn it has to corrspond to a singl arc (s p, t q ) that rflcts th xistnc of dg (p, q) in G. As a rsult, our fasibl ˆσ-flow f in Ĝ has to hav its cost ˆl(f ) to b at last b 1 2 and, furthrmor, ˆl(f ) b 1 2 is an uppr bound on th total amount of flow in f that is not transportd ovr th dirct on-arc flow paths (and thus passs through th vrtx v ). So, as w argud that th cost of f has to b at most b , thr is only at most 1 2 units of flow in f that passs through th vrtx v. Now, to xtract th dsird (fractional) nar-prfct b-matching x, w just tak x = f (sp,tq), for ach dg = (p, q) in G. Clarly, th siz of such fractional matching is at last b , which is wll abov our lowrbound of b 1 2 Õ(m 3 7 ) for a nar-prfct matching. Also, our construction works in Õ( m) tim, as dsird. Slack Variabls In th light of th abov, our goal now is to comput th nar-optimal solution to our minimumcost ˆσ-flow problm instanc in th graph Ĝ. Our approach to this task is inspird by so-calld path-following intrior-point mthods [Y97, Wri97, BV04]. At a vry high lvl, w will start with crtain initial solution that is far from bing optimal, and thn w will gradually improv in an 16

18 itrativ mannr its quality until clos-to-optimal solution is obtaind. This gradual improvmnt will b prformd in a vry spcific way. It will always try to push th currnt solution furthr down so-calld cntral path. Bfor w can dfin th cntral path, lt us first mntion that, in gnral, thr ar two natural ways of tracking th progrss of a currnt solution towards optimality. On of thm is purly primal and rlis on just maintaining a fasibl solution f and comparing its cost against som stimat of th cost of th optimal solution. Th scond on and th on that w will actually us hr is basd on primal-dual paradigm. Namly, in addition to maintaining a fasibl primal solution f, w will also kp a dual fasibl solution y. This dual solution provids an mbdding of all th vrtics in Ĝ into a lin, i.., y just assigns a ral numbr y v to ach vrtx v of Ĝ. Its fasibility condition is that for any arc = (v, w) of Ĝ it should b th cas that its slack variabl s := ˆl y w + y v is always non-ngativ, i.., that th lngth of th arc in this mbdding is nvr largr than its lngth according to th lngth vctor ˆl. Bfor w procd furthr, w not that th dual solution y is uniquly dtrmind up to a translation by th vctor s (givn th lngth vctor ˆl). So, for notational convninc, from now on, w will dscrib th dual solutions in trms of th vctor s instad of y. Duality Gap It is not hard to s that any fasibl dual solution s provids a lowr-bound on th cost of th optimal solution (aftr all, this is just a consqunc of wak duality). In particular, on has that for any pair (f, s) of fasibl primal and dual solutions, th so-calld duality gap, i.., th diffrnc btwn th uppr bound on th valu of optimal solution that is providd by th primal solution f and th lowr bound providd by th dual solution s is xactly f T s = µ T 1 = µ, whr µ := f s, for ach arc, and 1 is all-ons vctor (of dimnsion m). This mans that on can obtain a clos-to-optimal solution by dvising a procdur that (quickly) convrgs to a pair of primal and dual solutions (f, s) whos duality gap µ 1 is small (in our cas, at most 1 2 ). γ-cntrd Solutions and th Cntral Path To dscrib in mor dtail th convrgnc procss w will mploy, lt us associat with ach arc a masur ν 1. On can viw ν as a crtain notion of importanc of a givn arc. (Th motivation bhind introducing this notion will b clar latr.) W will always mak sur that th masurs of arcs ar not smallr than 1 and also that thir total sum is nvr too larg. That is, w will mak sur to maintain th following invariant. Invariant 5.2. W hav that ν T 1 = ν 4 m and for ach arc, ν 1. W want to not that whn discussing th prsrvation of th abov invariant w will only focus on nsuring that th upprbound is not violatd. Th fact that ν 1 for all arcs will b automatically nforcd as w will mak sur that th initial masur of all th arcs is always at last 1 and our algorithm will nvr dcras any masurs thy only might incras. 17